Question

The differential equation of the curve such that the initial ordinate of any tangent is equal to the corresponding subnormal is:

• A. a linear equation
• B. a homogeneous equation
• C. an equation with separable variables
• D. none of these

Answer 1

Answer: (A), (B)

a linear equation
a homogeneous equation

Equation of tangent $(x,y)$ to curve $y=f\left(x\right)$ is
$Y-y=f^{\prime }\left(x\right)(X-x)$
Putting $X=0$, the initial ordinate of the tangent is $y=x{f}^{\prime }\left(x\right)$
The subnormal at this point is given by $y\frac{dy}{dx},$ so we have
$y\frac{dy}{dx}=y-x\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{y}{x+y}$
$\Rightarrow \frac{dx}{dy}-\frac{x}{y}=1$
which is linear equation